Elementary Algebra
Law of Associativity
The law of associativity, also know as the associative property, is a law in mathematics applicable to the addition and/or multiplication of two groups of numbers that states the following symbolically:
$$ a + (b + c) = (a + b) + c \\and\\ a \times (b \times c) = (a \times b) \times c \\general\ form\\ a * (b * c) = (a * b) * c $$In general, the associative property state the grouping of factors in an operation can be changed without affecting the outcome of the equation. Thus a statement is said to be associate if it produces the same results regardless of the grouping of its operations. If the statement cannot satisfy the associative law, it is said to be non-associative meaning that changes to the groupings of operations will change the results.
List of Associative and Non-Associative Operations
Associative Operations
- Addition and multiplication of real numbers
- String Concatenation
- Addition and multiplication of complex numbers
- Addition and multiplication of quaternions
- Addition of octonions
- Calculation of Greatest Common Divisor (GCD)
- Calculation of Least Common Multiple (LCM)
- Union and intersection of sets
- Function composition for all mapping functions.
- Matrix multiplication
Non-associative Operations
- Addition and multiplication of floating-point numbers
- Subtraction and division of real numbers
- Multiplication of octonions
- Exponentiation
- Vector cross product
- Addition within a series
Law of Commutativity
The law of commutativity, or the commutative property is similar to the associative property but rather than the grouping effecting the result, the commutative property examines the ordering of the operators in a binary operation.
Point-Slope Formula
$$ (y - y_1) = ((y_2 - y_1) / (x_2 - x_1)) * (x - x_1) $$Straight Line Formula
This equation is derived from the point-slope formula
$$ (y - y_1) * (x_2 - x_1) = (x - x_1) * (y_2 - y_1) $$